In this talk I will review and address the long standing problem of determination of the state of matter in the cores of neutron stars. Our understanding of the cold, dense nuclear matter in neutron star interiors has suffered a dramatic revolution during the recent years. On the one hand, laboratory experiments have been able to probe higher and higher densities that comprise the equation of state (EoS) of dense nuclear matter. On the other hand, multi-messenger astronomy observations have brought new physical constraints that narrow the parameter space of the different EoS models. There is however, certain tension between astrophysical and terrestrial measurements. One practical way to implement all the available constraints is implementing a Bayesian analysis for model comparison, which I will present.
In order address the corresponding to state-of-the-art physics, I shall introduce different approaches to the neutron star EoS: pure hadronic stars, stars with hyperonic and condensate content, and hybrid stars whose cores supports deconfined quark matter as well as strange quark stars, for the sake of comparison. Moreover, I will emphasize the role of the nuclear symmetry energy in determination of the star properties, discuss the hyperon puzzle and present a particular model of hybrid stars: mass twins.